_{1}

In this paper, we present a theoretical investigation as a function of temperature of a critical anomaly find in InAlAs hetero-structure of two-dimensional electron gas. This study has shown the presence of a large and continuous anomaly. This anomaly is explained through a theory based on the general assumption. The present theoretical research is based essentially on the characteristic of specific heat capacity extending over a large temperature, but we underline a good agreement with results of the relation with chemical potential and Broadening parameter as a function of temperature. It is found that the specific heat capacity observed by a peak at low temperature, at a critical temperature, is directly linked to Schottky anomaly and unveiling the existence of phase transition in InAlAs. Our results are completed by the study of the dependence of the heat capacity on the spin as a function of temperature. This study confirms the same behavior with result without spin.

Electron in a Two-dimensional gas (2DEG) continues to be of interest in physics since they exhibit non classical behavior and are readily realizable in semiconductor hetero-structure [

On the side of theorical investigation, the general trend is attributed to the presence of Landeau levels. These Landeau levels are indicated by broadened energy levels, characterized by phenomenological broadening parameter Γ, which is represented in the density of states (DOS) [_{n}. These E_{n}, known as Landeau levels, are the quantized energy spectrum (E) obtained when a strong electric field (V) is applied to the system’s plane [

These striking behavior of E_{n} broadening, from which other exotic features arises, has been attributed to disorder due to the presence of impurities, defects and other inhomogeneity’s in the system [

In our work, we present a numerical calculation of a new specific heat capacity S_{heat} anomaly, characterized of InAlAs devices, for different chemical potential µ, as a function of temperature T. The purpose of the present papers is to give furthers study for 2DEG. We show the presence of a large and continous anomaly at low temperature with the increase of chemical potential. Then we finish by a summary, that we have compared our results with other references.

During the early part of the nineteenth century, studies on the heat capacity of materials tended to indicate that it was rather uninteresting property somewhat independent of temperatures. Heat capacity is one of the most fundamental physical properties as it directly probes thermodynamic quantities such as entropy [

Thermodynamic properties of each device that can be described by the independent or free particle model are determined by their single particle energy spectra depending on the confinement and the size of the system.

For such system, the specific heat capacity of a substance S_{heat} of 2DEG was studied as a function of temperature T [_{heat}, in the Debye model [

Specific heat capacity is one of the fundamental thermodynamic properties of a substance. It is defined as the energy that has to be transferred to or from a unit of mass or amount of substance to change the system temperature by one degree. The heat capacity of material is a property that indicates the amount of thermal energy the material must absorb to achieve a specified temperature rise. Specific heat capacity is generally sensitive to phase change. Different materials of a given mass require different quantities of heat to rise their temperature by specified value since different materials absorb energy in different ways. Specified heat capacity of material is obtained from expression:

S heat = ∂ U ( T ) ∂ T = ∂ ∂ T ∫ − ∞ + ∞ f ( E , μ , T ) ⋅ ( E − μ ) ⋅ D ( E ) d E (1)

where μ the chemical potential, D ( E ) is the density of states, and f ( E , μ , T ) is the Fermi Dirac distribution function giving by:

f ( E , μ , T ) = 1 1 + exp ( E − μ K B T ) (2)

where K B is the Boltzmann’s constant, T is the absolute temperature, E is the energy of the single particle state, μ is the chemical potential and D ( E ) is the density of states DOS. The temperature derivation in Equation (1) then acts only on f ( E , μ , T ) . S heat behavior of can be determined once the density of electron N and the chemical potential μ are known.

where:

N = ∫ − ∞ + ∞ f ( E , μ , T ) ⋅ D ( E ) d E (3)

To calculate the density of electron at each Landau level we need to know the density of states function for electrons.

Ideally, the DOS of a non-interacting 2DEG is given as a series of delta function [

D ( E ) = D 0 ∑ n δ ( E − E n ) (4)

where D 0 is a constant which depend on devices mass, And E n = h 2 K 2 2 m , where K = n π a , is the energy of the n^{th} Landeau level. Such a DOS structure can be used to model actual materials with a very narrow distribution of energy carrier [

Usually, the actual shape of the density of states of a 2DEG is determined by making theoretical fits to the heat capacity data from experimental measurements. There are many form of the DOS used in literature [

D ( E ) = D 0 ∑ n 1 2 π Γ exp ( − ( E − E n ) 2 2 Γ 2 ) (5)

where, Γ is the broadening is taken into account by the parameter.

When the chemical potential is temperature dependent [

Evaluating the temperature derivative in Equation (1) results into:

S heat = ∫ − ∞ + ∞ ∂ f ( E , μ , T ) ∂ T ⋅ ( E − μ ) ⋅ D ( E ) d E − ∫ − ∞ + ∞ f ( E , μ , T ) ⋅ ∂ μ ∂ T ⋅ D ( E ) d E (6)

Using in this expression the derivative equation of the Fermi function, we can obtain the general equation for the specific heat capacity.

In the Framework of the effective mass approximation in two-dimensional electron gas system, the Schrödinger equation in the effective mass is giving by:

H ψ ( x , y , z ) = E ψ ( x , y , z ) (7)

With Hamiltonian subjected with an electrical potential, in a Cartesian system, written in this form:

H = − ℏ 2 2 m * Δ + V ( z ) (8)

where V ( z ) is the self-consistently calculated potential energy which includes contribution arising both from dopant charges and electrons localized in the quantum well and it is expressed as V ( z ) = e c A → , m * is the effective mass, ℏ is Planck’s constant divided by 2π, and E is the energy eigenvalue.

By computing the solution of the Schrödinger equation, we can assuming that the Landeau levels, of the n^{th} energy level is E n = h 2 K 2 2 m . This result is clearly represented by the different states of energy for different point in the band gap (

Assuming a Gaussian distribution of defect states in the gap, broad distribution of the density of states was found in InAlAs devices. This result can be represented by

The obtained distribution of density of states can now be used to calculate the temperature dependences of the heat capacity in a wide temperature range, for the case when the chemical potential µ is varied between 0 meV to 0.7 meV in 2 DEG semiconductor III-V, and with a fixed value of parameters Γ = 0.4 meV. This result was represented by _{heat} exhibit a single peak at the low temperature (as it is clearly shown in the insert of _{heat} is maximum, T_{peak}, is equal to (12.7 ± 1.2) K. Let’s us note that the peak temperatures of the heat capacity are shifting with chemical potential μ. This is expected, since we are computing the Specific heat capacity. If we computed the total heat capacity of the system, these peaks would occur at the same temperature.

S_{heat} increase with the increasing of T, then display a sharp peak at T_{peak}, before the decreasing at high T. The maximum values occurs at a temperature very closed to that shown in this figure indicating that the observed features are likely related to Schottky behavior. This behavior happens when the heat capacity of the nuclei is comparable to that of the electron.

While the temperature at which the peak in S_{heat} appears is about (12.7 ± 1.2) K, there is no charge of the peak temperature for a further increase of chemical potential μ. Another point in support of this is the observation of the broadening of the peak become strengthen when μ increase; add the width of the peak become more large with the μ decrease.

The peak in S_{heat} vs. T observed at low temperature is dependent of µ. Whereas the insert of _{c} strongly depends on μ. The origin of the peak at very low temperature T is discussed in relation with a phase transition in the electronic system.

This phenomenon is also observed in the same device but at another value of parameter Γ equal to 0.1 meV (

These results obtained at Γ = 0.1 meV are in good agreement with those at Γ = 0.4meV, with a monotonous increase of S_{heat} with the increasing of temperature. This exothermic peak proves the presence of anomaly behavior in InAlAs. Taking into account the thermodynamic properties and structural measurement, we propose that the presence of this anomaly is clarified on the basis of the idea of the Schottky anomaly.

The presence of peaks at different value of the chemical potential µ, and in a wide temperature range, correspond to anomaly is an observed effect where, the specific heat capacity shows an exothermic peak. These anomalies are likely related to Schottky anomaly behavior which confirms the presence of defect in semiconductor. The shape and the sharpness of the peaks are suggestive of a phase transition in InAlAs, which we propose that they are in reality states of trapped defect [

Specific heat capacity for a fixed value of chemical potential (µ = 0.5 meV ) and at two different broadening parameter Γ, one equal to 0.1 meV and the other fixed at 0.4 meV, are compared in

Now, we discuss the relation between specific heat capacity S_{heat} vs. critical temperature T_{c} of the maximum of peaks observed at low temperature in our devices, at 0.1 meV and at 0.4 meV parameters Γ, and for a large scale of chemical potential µ varying between 0 meV to 0.7 meV. The relation between heat capacity and critical temperature are studied by different auteurs [

Specific heat capacity is linear proportional to the maximum of peak temperature as shown in _{heat} peak with the increasing of temperature is clearly visible. The magnitude of specific heat capacity jumps at the transition temperature T_{c}.

The strong and non-monotonic dependence of T_{c} on chemical potential µ (

The linear decrease of critical temperature T_{c} with the increase of chemical potential μ in our device is confirmed in

To complete the discussion on the heat capacity, let us discuss the dependence of the heat capacity on the spin size on the InAlAs devices as a function of temperature [

Notice that with increasing of spin size the peak structure shifts to higher temperature due to the increased interaction energy that leads to a larger gap in the energy spectrum. Also, the insert of

We have studied the cardinal behavior of the specific heat capacity S_{heat} of two-dimensional electron gas (2DEG) systems which are performed on InAlAs hetero-structures electron layers. Specific heat capacity has been simulated for different chemical potential and for different broadening parameter. This study exhibits remarkable peaks temperatures, which are shifting for different parameters varying in our studies. The magnitude of the specific heat capacity jumps to the critical temperature T_{c} and the exponentially vanishing specific heat at low

temperature, unveiling the existence of phase transition in InAlAs device, occurring the 2DEG. We suggest that this anomaly deduced from the specific heat capacity is directly related to Schottky anomaly. The study of critical temperature T_{c}, is also made with different chemical potential μ and at 0.4 meV and 0.1 meV broadening parameter Γ. Our data underscore the importance of the coupling between the 2DEG and spins. These results, with spin, have the same behavior without spin.

The author declares no conflicts of interest regarding the publication of this paper.

Bouzgarrou, S. (2020) New Anomaly at Low Temperature for Heat Capacity. Open Access Library Journal, 7: e6477. https://doi.org/10.4236/oalib.1106477